Solve 1418519125=106^x

$$1418519125= { 106 }^{ x }$$

Solve for x

$$x=\log_{106}\left(1418519125\right)\approx 4.518742256$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$106^{x}=1418519125$$

Take the logarithm of both sides of the equation.

$$\log(106^{x})=\log(1418519125)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$x\log(106)=\log(1418519125)$$

Divide both sides by \log(106).

$$x=\frac{\log(1418519125)}{\log(106)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$x=\log_{106}\left(1418519125\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(106)}+\log_{106}\left(1418519125\right)$$

$$n_{1}\in \mathrm{Z}$$